Fractions as approximations

The other day I heard the business reporter on the radio say "Advancing stocks led declining stocks by 7 to 5." That is, 7/12 of all stocks went up for the day, and 5/12 went down.

Now, there are thousands of stocks in the market, so 7/12 is most likely just an approximation to the actual number. My question is, how good an approximation is it? You have enough information to answer the question if you make one reasonable assumption.

The other day I heard the business reporter on the radio say "Advancing stocks led declining stocks by 7 to 5." That is, 7/12 of all stocks went up for the day, and 5/12 went down.

Now, there are thousands of stocks in the market, so 7/12 is most likely just an approximation to the actual number. My question is, how good an approximation is it? You have enough information to answer the question if you make one reasonable assumption.

You can take the division symbol to indicate perfect accuracy, so any errors in the final answer are due entirely to the numbers 7 and 12. So the question becomes how good an approximation are 7 and 12?

The other day I heard the business reporter on the radio say "Advancing stocks led declining stocks by 7 to 5." That is, 7/12 of all stocks went up for the day, and 5/12 went down.

Now, there are thousands of stocks in the market, so 7/12 is most likely just an approximation to the actual number. My question is, how good an approximation is it? You have enough information to answer the question if you make one reasonable assumption.

If there is an infinite population of stocks, and you sample N of which n when
up, and estimate proportion P of the pouulation that went up by p=n/N.
Then if N is large enough n is a random variable with a binomial distribution
with mean

mu = NP

and standard deviation:

sigma = sqrt(NP(1-P))

so p is a RV with SD sigma/N.

If instead we look at the entire population of size N, and find n went up then
there is no error in putting p=n/N as the proportion that went up.

Hmm, not going in the direction I intended so let me clarify a little.

Given a real number (not necessarily rational) between 0 and 1, you want to find the "best" rational approximation to it. You might say for example "half of the voters in the US are Democrats." Now the actual proportion is unlikely to be exactly .50000..., but maybe it's really .49653..., so most people would say 1/2 is the best rational approximation you're going to come up with.

So my original question can be restated, If 7/12 is the best rational approximation to a particular real number, what can you tell me about that number? How good an approximation is 7/12? There's already enough information to solve the problem if you make one reasonable assumption.

The other day I heard the business reporter on the radio say "Advancing stocks led declining stocks by 7 to 5." That is, 7/12 of all stocks went up for the day, and 5/12 went down.

Now, there are thousands of stocks in the market, so 7/12 is most likely just an approximation to the actual number. My question is, how good an approximation is it? You have enough information to answer the question if you make one reasonable assumption.

well, it would be hard to tell exactly how accurate this ratio is without actually knowing the figures. How many stocks are they talking about? We don't know. It does, however, seem like they are going by ratios (for every 12 stocks counted, 7 went up and 5 went down), so I don't expect it to be inaccurate enough to cause concern.

Hmm, not going in the direction I intended so let me clarify a little.

Given a real number (not necessarily rational) between 0 and 1, you want to find the "best" rational approximation to it. You might say for example "half of the voters in the US are Democrats." Now the actual proportion is unlikely to be exactly .50000..., but maybe it's really .49653..., so most people would say 1/2 is the best rational approximation you're going to come up with.
.

This is one of my favorited things I studied. The best possible rational approximation* of an irrational number occurs as the continued fraction for the number.

*)We define it the most accurate** rational with a given smallest denominator

There is no way to get an irrational number from the problems you presented.

Let's say that 33 out of 100 random people were guys. That would mean that the percentage of guys is 33.33333333333333333333333333333333333333333333333 333333333333333333333333333333333
or you could just say that it's 2/3 guys.

This is one of my favorited things I studied. The best possible rational approximation* of an irrational number occurs as the continued fraction for the number.

*)We define it the most accurate** rational with a given smallest denominator

**)We define it that |x-p/q|<e where e is the error.

To use a colloquialism, isn't this arse about face, the continued fraction givesthe best rational approximation to an irrational number, but here we are talking about being given a rational approximation to a (not-necessarily irrational number - in fact it has to be rational as there are a finite number of stocks) what range of proportions are consistent with that rational approximation.

In these terms we see that there is in fact no answer because we don't knowwhat psychological factors determined how the number quoted was derived,i.e. what the rounder/approximater considered an socially acceptable rounded proportion.

In everyday life, if somebody uses 7/12 as an approximation, it must be because there is no fraction with a lower denominator that gives a better approximation.

That means the number is closer to 7/12 (=.58333...) than it is to 4/7 (=.57142...) or 3/5 (=.6). So the number must be between .57738... and .59166..., or an error of less than 1% in either direction.